Agree with the others. The Vatican is overwhelming and I went there on my second (full) day in the city--two days after landing. I think you can do St. Peters' and certainly enjoy strolling around the Piazza and the neighborhood. We stayed in that area and I really enjoyed just walking around, especially the very first day.
If, due to time constraints, you absolutely have to do the Vatican the first day--and you plan to book a tour--I would say book the afternoon (usually around 1pm or 2pm) tour. That way if your plane is delayed or you are tired and need a nap, you aren't hemmed in by a morning deadline. But I wouldn't expect to get as much out of it as you probably will when rested.
For a realistic time estimate: My plane landed just before 7am on a Sunday morning. I, with a small carryon and personal bag, was out of the airport in about 30 minutes. The train takes about another 30 minutes. Termini was confusing and by the time I found the metro and bought a ticket, that was probably another 15-20 minutes. If it's a weekday you will be on the bus in rush hour. Depending on where specifically your B&B is, I would suggest you consider taking the metro as opposed to the bus--especially the #64, notorious for pickpockets.
Then it depends on how long it takes your B&B to have your room ready, check in, etc. Our hotel didn't have the rooms ready but let me leave my bags in a back room while I went exploring.
For the Borghese--I would suggest booking the 3pm or 5pm slot. That way you can do your Scavi tour, then rest and/or have lunch afterwards, and then make your way over to Villa Borghese (which takes a little while to get to from the Vatican area).